Badiou, Extension, and Networks

One of the most attractive, problematic, and astonishing features of Badiou’s ontology is his strictly extensional understanding of sets or multiplicities. A set is not defined by its members sharing a common predicate or quality, nor by the relations among members of the set. Rather, a set is defined strictly by its extension or the members that belong to that set. From the standpoint of 20th Century French and German Continental philosophy, this thesis cannot but be a heresy, for the predominant trend in Continental thought has been a relational conception of entities. Whether we are speaking of language as a diacritical set of negative oppositions as defended by the structuralists and the post-structuralists, or Heidegger’s being-in-the-world where entities, the ready-to-hand, are defined by the relational networks to which they belong, the predominant trend has been to treat beings as bundles of relations such that the entity is nothing apart from its relations. In a spirit similar to Deleuze’s declaration that relations are always external to their terms, Badiou will have none of this. For Badiou entities are not defined by their relations and there are no intrinsic or internal relations that define the being of the entity. Rather, they are simply defined by their relations.

From the standpoint of both Heidegger’s being-in-the-world where each entity is thought as a “being-in” belonging to the worldhood of the world defined by an ensemble of relations defining meaning, or from the standpoint of structuralist and post-structuralist thought where the entity is an ensemble of internal relations from which it cannot be detached, or from the standpoint of Hegelianism where, as Hegel painstakingly shows in the Doctrine of Essence in the Science of Logic, where the entity simply is its relations or mediations, this move cannot but appear stunning. For what this extensionalist conception of sets authorizes is combinations of subsets in whatever order we might like. This, in short, is what the axiom of union tells us. What the axiom of union allows– if I understand it correctly (I’m sure Dominic will educate me if I don’t, thankfully) –is the construction of whatever sets we might like based on those elements belonging to our initial set. Thus, if I have a set composed of an umbrella, an apple, and the moon ({umbrella, apple, moon}), I certainly have a set composed of the apple and the moon ({apple, moon}), or a set composed simply of the apple ({apple}).

Now all of this sounds silly and unremarkable so long as we don’t contrast Badiou’s extensional notion of sets with the relational ontologies that have predominated during the 19th and the 20th century. If to be an entity is to be a bundle of internal relations, it follows that entities cannot be grouped in any way we might like. Rather, a model of the world based on internal relations dictates that each entity necessarily has a place within an Order and that the entity is nothing apart from this order. Thus the phoneme {c} is nothing apart from other phonemes such as {p}, {b}, {f}, etc., by virtue of the differentiality that allows it produce different senses at the level of the signifier: cat, pat, bat, fat. Insofar as these phonemes take on their value (in the linguistic sense of “value”) differentially in relation to one another, they are nothing independent of their relations to one another. This is what it means to say that each entity takes on a place within an Order. The Order is the totality of internal relations defining a system or structure, whereas the places are locations within that Order relative to the other terms. Because the relations are internal to the various beings in the Order, there is thus a Law that governs these beings and exhausts their being, legislating how they can and cannot act.

In proposing that sets are defined purely by their extension or their membership, Badiou undermines the thesis that to be is to be a bundle of internal relations. At the level of ontology, there is thus no intrinsic Order that defines entities. Rather, in their stark independence, the elements that make up a set not only can be decomposed into infinite subsets (through a recursive process of taking the power set of each power set), but the elements of each set can be related in a variety of different was or simply taken as singletons, thereby abolishing the notion of intrinsic or internal relations as in the case of Hegel’s logic of essence.

read on!

Of course, while this might be the case ontologically or in terms of what can be said of being qua being independent of any particular being, worlds, the domain of the ontic, do not work in this way. In worlds it is not the case that things can be combined in any old way, but rather there are ordering relations among entities. However, it’s notable that Badiou’s thesis is not that entities cannot enter into relations, only that they are not defined by their relations. Thus, while there is a gap between the ontological (being qua being or the discourse of multiplicity qua multiplicity) and the ontic (ordered or related elements in a world), Badiou’s ontology nonetheless strongly suggests that entities are prior to their relations in the domain of the ontic as well. Here relations are external to their terms, such that entities are not defined by their relations, but rather enter into their relations.

If entities are independent or prior to their relations, it follows that we should abandon the concept of structure and instead shift to a network or assemblage based model of relations among entities. The problem with the concept of structure is that it treats relations as internal relations, such that the elements belonging to the structure have no existence independent of its relations. In his marvelous popularization of network science, the Columbia sociologist Duncan Watts admirable puts his finger on this problem, writing that,

The crux of the matter is that in the past, networks have been viewed as objects of pure structure whose properties are fixed in time. Neither of these assumptions could be further from the truth. First, real networks represent populations of individual components that are actually doing something— generating power, sending data, or even making decisions. Although the structure of the relationships between a network’s components is interesting, it is important principally because it affects either their individual behavior or the behavior of the system as a whole. Second, networks are dynamic objects not just because things happen in networked systems, but because the networks themselves are evolving and changing in time, driven by activities or decisions of those very components. In the connected age, therefore, what happens and how it happens depend on the network. And the network in turn depends on what has happened previously. It is this view of a network as an integral part of a continuously evolving and self-constituting system– that is truly new about the science of networks. (Six Degrees, 28 – 29)

Structuralism has been valuable in drawing our attention to the importance of relations. However, the cardinal sin of structuralist thought– and these assumptions still remain pervasive today –lies in its tendency to, as Bateson would put it, confuse the map with the territory. That is, the structuralist makes a map of relations among nodes in a network, but then treats this map as if it were itself a real and abiding thing such that these relations are abiding and eternal. The entities inhabiting the network then get treated as epiphenomena of this map of relations, such that it is the structure that is real and the entities populating the network that are illusions. This is a specifically Platonic tendency within structuralist thought.

In ontologizing structure in this way, the dynamics of structure through which structure is both produced and reproduced in time. In short, what is missed is the manner in which nodes in a network must be related. That is, the links among elements of a network must be forged for the network to function. Part of the great value of structuralism has been to draw our attention to the manner in which there are emergent properties of networks that exceed the intentions of any of those participating in the network (for example, patterns of wealth distribution). However, by ignoring the dynamics of networks and the fact that they have to be built, structuralists have drawn the wrong conclusion. Thus, for example, Althusser drew the conclusion that humanism must be mistaken as these networks function anonymously and not according to the intentions of those participating in the structure. The individual person thus becomes, under this reading, a sort of illusion and nothing more than its place in the social structure.

Althusser, however, is wrong on both counts. On the one hand, insofar as entities are prior to their relations, they are not simply illusions (though their effect might be negligible from the standpoint of the functioning of the network). Moreover, without the interactions among these individuals, the network could not exist at all. Thus, while a network cannot be reduced to the action of these individuals, it also can’t exist without the actions of these individuals forging links, making decision, becoming hubs, and therefore attracting more relations that then come to preside over the future course of the networks development. This last point is especially important. One thing network research has discovered is that those nodes in a network that possess more relations to other nodes within a network also attract more relations as the network evolves and develops. For example, wealth tends to attract more wealth such that it comes to be localized in one segment of the population. Thus, during the early stages of network development, the relations that can be forged among entities are relatively open. But, networks are defined by times arrow such that the forging of relations introduces elements of accretion that limit the direction in which the network can develop in the future.

From the standpoint of political theory, this simple observation is of tremendous importance. First, it underlines a point of strategy for targeting oppressive social systems in that a network will be weakest at those points where relations to a particular node or set of nodes are most extensive. Take out that node and the rest begins to fall (as we have learned from the California power outages). Second, it also underlines the importance of developing group relations in engaging a network and changing it. In other words, it is of vital importance to generate networked relations that will attract more relations to other nodes if the overall evolution or development of a network is to be affected in a significant way. This shows, for example, why forms of political theory written in such a way to interrupt discourse and communication so as to fight the metaphysico-politico structure in language itself are so misguided. By adopting this rhetorical strategy they limit the ability for links among nodes to be formed, thereby preventing the accumulation of relations within networks that are the best chance for shifting the organization of the network as a whole.

On the other hand, if Althusser’s anti-humanism is mistaken it is because it treats the individual within a structure or network as a sort of illusion or effect of the structure. Althusser’s point is well taken. There are emergent properties of networks that can’t be reduced to the intentions of the individuals caught like a fly within these networks. However, this is very different than the conclusion that the individual is nothing but an effect of its place within a network. First, individuals are prior to their relations, and as such cannot be reduced to their relations. Indeed, in the world nothing ever functions as smoothly as our maps of networks suggest (Bourdieu analyzed this point to great effect in his critique of structuralist models of kinship relations in The Logic of Practice). Second, individuals move among different networks that are discontinuous to one another, thereby indicating that they are irreducible to their relations. This does not mean that the ongoing relations among elements in a network don’t play a tremendous constraining role on individuals participating (whether or not they know it) in a network; but to point this out is different than claiming that the individual is its place in a network.

Ontology matters. As Mark of K-Punk so beautifully put it recently,

My instinct would be to reverse this, i.e. it’s not that ontology is always constructed through a political battle, but that politics is always constructed through an ontological battle. Politics certainly presuppose ontology – to take a glaring example, the key slogans of Thatcherite capitalist realism, for instance (“There is no such thing as society, only individuals and their families” and “There is no alternative”) were explicitly ontological claims, claims about what sort of entities can be said to exist in the world. But that isn’t to say that all ontologies presuppose a politics.

This tendency to ignore the dynamics of networks, the fact that they must be produced and reproduced, the fact that relations among nodes in a network are perpetually shifting and changing, leads to the wrong sorts of questions and problems at the level of political theory. Thus, for example, Žižek has written thousands of pages arguing that we must conceive of a split subject, a subject that is a gap within structure, in order to account for how change is possible. However, this thesis only rings true if we begin from the premise that individuals are effects of structure without any independence of their own. All of this changes when we no longer ontologize structure as a Platonic form presiding over entities and reducing them to pure vehicles of structure. Rather, the question instead becomes that of how links between nodes are formed, what those links are, where the hubs are, and how it might become possible to form new networks. The question of change no longer looks so mysterious or intractable.